A 0-1 Law for Multifractal Spectra
via the HGDS Scale Derivative
January 01, 1970
We prove that the multifractal spectrum \(D(h,\omega)\) of a stochastic process is almost surely deterministic under a scale decorrelation condition on the HGDS scale derivative \(\eth_\Lambda X\). The key difficulty is that the pointwise Hölder exponent lives in the germ \(\sigma\)-algebra, where classical 0-1 laws do not reach. We get around this by working with the geometry accumulation integral \(G_\Lambda\), which is a genuine Lebesgue integral over scales and concentrates almost surely. The boundary case — log-correlated fields — is sharp: the variance summability condition fails exactly there.
Let \(X:[0,1]\times\Omega\to\mathbb{R}\) be a stochastic process. The pointwise Hölder exponent at \(t\) is \[h_X(t,\omega) = \liminf_{r\to0}\frac{\log|X(t+r,\omega)-X(t,\omega)|}{\log|r|},\] and the multifractal spectrum is \(D(h,\omega)=\dim_H\{t\in[0,1]:h_X(t,\omega)=h\}\). For most natural processes — fractional Brownian motion, Lévy processes — the spectrum is the same for almost every sample path. But there is no general theorem that says when this happens. The question of finding a condition \(C(X)\) such that \[C(X)\implies D(h,\omega)=\bar D(h)\quad\text{a.s.}\] is completely open [1].
The obvious approach is to use Kolmogorov or Hewitt-Savage. Both fail for the same reason: \(h_X(t,\omega)\) is defined by a lim inf as \(r\to0\), so it lives in the germ \(\sigma\)-algebra at \(t\). Kolmogorov’s law needs tail events. Hewitt-Savage needs exchangeability. Neither is available here.
The HGDS framework [2]–[4] splits increments at each scale \(\Lambda>0\): \[X(t+\Lambda,\omega)-X(t,\omega) = \eth_\Lambda X(t,\omega)\cdot\Lambda + R_\Lambda X(t,\omega),\] where \(\eth_\Lambda X\) is the macro scale derivative (from wavelet truncation) and \(R_\Lambda X\) is the fine-scale residual. Instead of working with the lim inf directly, we work with \[G_\Lambda(t,\omega) = \int_\Lambda^1|\eth_s X(t,\omega)|\cdot s^{-1}\,ds,\] which satisfies \(h_X(t,\omega)=1+\lim_{\Lambda\to0}\log G_\Lambda(t,\omega)/ \log\Lambda\). Since \(G_\Lambda\) is a Lebesgue integral over scales rather than a lim inf, it is accessible to concentration arguments.
We say \(X\) has locally stationary increments if for each \(t\in[0,1]\), the map \(t\mapsto\mathbb{E}[f(X(t+r)-X(t))]\) is differentiable for every bounded continuous \(f\) and every \(r>0\). This is slightly weaker than stationary increments and covers the genuinely multifractal case where \(\bar{h}(t)\) varies.
Theorem 1 (0-1 Law). Let \(X:[0,1]\times\Omega\to\mathbb{R}\) have locally stationary increments, be jointly measurable, and have HGDS macro derivative \(\eth_\Lambda X\) defined via wavelet decomposition with parameter \(H\in(0,1)\). Write \(\bar{h}(t)=\mathbb{E}[h_X(t,\cdot)]\) and \(\bar D(h)=\dim_H\{t:\bar{h}(t)=h\}\).
Suppose: \[\begin{align} &\sum_{n=1}^\infty \frac{\int_{e^{-n}}^1\!\int_{e^{-n}}^1 |\mathrm{Cov}(\eth_s X(t),\eth_u X(t))|\,(su)^{-1}\,ds\,du}{\bigl(\int_{e^{-n}}^1\mathbb{E}[|\eth_s X(t)|]s^{-1}\,ds\bigr)^2} <\infty\quad\text{uniformly in }t,\\ &\mathbb{E}\bigl[|\eth_s X(t)-\eth_s X(t')|^p\bigr] \le C|t-t'|^{pH}s^{p(H-1)} \;\text{ for some }p>2/H,\\ &\mathbb{E}\bigl[|\eth_s X(t)|\bigr]\asymp s^{\bar{h}(t)-1} \text{ as }s\to0,\;\text{uniformly in }t,\\ &\bar{h}(t)\text{ is not constant on any interval.}\end{align}\] Then \(D(h,\omega)=\bar D(h)\) a.s.for all \(h\ge0\).
Remark 2. Conditions (C-i)–(C-iv) hold for fractional Brownian motion, Lévy stable processes with finite variance, and standard multifractal cascades. Log-correlated fields fail (C-i) — this is sharp; see Section 6.
The condition \(p>2/H\) in (C-ii) (rather than \(p>1/H\)) is needed for Kolmogorov’s continuity criterion in the proof of Lemma 3. For \(H\le1/2\) this is automatic from \(p>1/H\) since \(1/H\ge2\). For \(H>1/2\) the stronger condition is genuinely required.
Proposition 3. \(\omega\mapsto D(h,\omega)\) is \((\mathcal{F},\mathcal{B}(\mathbb{R}))\)-measurable for each fixed \(h\ge0\).
Proof. Write \(h_X(t,\omega)=\sup_{n\in\mathbb{N}}\inf_{r\in\mathbb{Q}\cap(0,1/n)} \frac{\log|X(t+r,\omega)-X(t,\omega)|}{\log|r|}\), a countable sup/inf of measurable functions, jointly measurable in \((t,\omega)\).
Set \(A=\{(t,\omega):h_X(t,\omega)=h\}\in\mathcal{B}([0,1])\otimes\mathcal{F}\). For open \(U\subseteq[0,1]\): \(\{\omega:E(h,\omega)\cap U\ne\emptyset\}=\pi_\Omega(A\cap(U\times\Omega))\) is the projection of a Borel set, hence analytic, hence measurable by completeness of \((\Omega,\mathcal{F},\mathbb{P})\).
The map \(\omega\mapsto\mathcal{H}^s_\delta(E(h,\omega))\) is measurable as a countable infimum over rational covers. Passing \(\delta\to0\) and using \(\{\omega:\dim_H E(h,\omega)<c\}=\bigcup_{s\in\mathbb{Q}\cap(0,c)} \{\omega:\mathcal{H}^s(E(h,\omega))=0\}\) gives the result. 0◻ ◻
Write \(\tilde{G}_\Lambda=G_\Lambda/\mathbb{E}[G_\Lambda]\).
Lemma 1. Under (C-i), \(\sum_{n=1}^\infty\mathrm{Var}(\tilde{G}_{e^{-n}}(t))<\infty\) for all \(t\).
Proof. Direct computation: \[\mathrm{Var}(\tilde{G}_\Lambda(t)) = \frac{\int_\Lambda^1\!\int_\Lambda^1 |\mathrm{Cov}(\eth_s X(t),\eth_u X(t))|\,(su)^{-1}\,ds\,du}{\bigl(\int_\Lambda^1\mathbb{E}[|\eth_s X(t)|]s^{-1}\,ds\bigr)^2},\] which is the summand in (C-i) at \(\Lambda=e^{-n}\). 0◻ ◻
Lemma 2. Under (C-i), \(\tilde{G}_{e^{-n}}(t,\omega)\to1\) a.s.for each fixed \(t\).
Proof. Chebyshev gives \(\sum_n\mathbb{P}(|\tilde{G}_{e^{-n}}(t)-1|>\varepsilon) \le\varepsilon^{-2}\sum_n\mathrm{Var}(\tilde{G}_{e^{-n}}(t))<\infty\) by Lemma 1. Borel-Cantelli and monotonicity of \(G_\Lambda\) in \(\Lambda\) complete the proof. 0◻ ◻
Lemma 3. Under (C-ii) with \(p>2/H\), \(t\mapsto G_\Lambda(t,\omega)\) is a.s. \(\beta\)-Hölder with \(\beta=H-1/p>0\): \[\mathbb{E}\bigl[|G_\Lambda(t)-G_\Lambda(t')|^p\bigr]^{1/p} \le C|t-t'|^H\Lambda^{H-1}.\]
Proof. By Minkowski’s inequality in \(L^p\), then (C-ii): \[\begin{align} \mathbb{E}\bigl[|G_\Lambda(t)-G_\Lambda(t')|^p\bigr]^{1/p} &\le\int_\Lambda^1\mathbb{E}\bigl[|\eth_s X(t)-\eth_s X(t')|^p\bigr]^{1/p} s^{-1}\,ds\\ &\le C|t-t'|^H\int_\Lambda^1 s^{H-2}\,ds = C|t-t'|^H\cdot\frac{\Lambda^{H-1}-1}{1-H}\\ &\le C'|t-t'|^H\Lambda^{H-1}. \end{align}\] Kolmogorov’s continuity criterion applies: the Hölder exponent is \(\beta=H-1/p>0\), and \(p\beta=pH-1>1\) iff \(p>2/H\), which holds by (C-ii). 0◻ ◻
Proposition 4. Under (C-i) and (C-ii): \(\sup_{t\in[0,1]}|\tilde{G}_{e^{-n}}(t,\omega)-1| \to0\) a.s.
Proof. Fix \(\varepsilon>0\). Cover \([0,1]\) by \(M=\lceil\delta^{-1}\rceil\) points \(\{t_i\}\) with spacing \(\delta\).
Grid points. \(\mathbb{P}(\max_i|\tilde{G}_{\Lambda_n}(t_i)-1|>\varepsilon) \le M\varepsilon^{-2}\sup_t\mathrm{Var}(\tilde{G}_{\Lambda_n})\).
Between grid points. By Lemma 3, \(|\tilde{G}_{\Lambda_n}(t)-\tilde{G}_{\Lambda_n}(t_i)| \le C(\omega)\delta^\beta\).
Combine. Set \(\delta=(\varepsilon/2C)^{1/\beta}\) so \(M\lesssim\varepsilon^{-1/\beta}\): \[\mathbb{P}\!\left(\sup_t|\tilde{G}_{\Lambda_n}(t)-1|>2\varepsilon\right) \le\frac{C'\sup_t\mathrm{Var}(\tilde{G}_{\Lambda_n})}{\varepsilon^{2+1/\beta}}.\] Sum over \(n\) and apply Borel-Cantelli using (C-i). 0◻ ◻
Lemma 4. Under locally stationary increments and (C-i) and (C-iii), \(h_X(t,\omega)=\bar{h}(t)\) a.s.for each fixed \(t\).
Proof. By Lemma 2, \(\tilde{G}_{e^{-n}}(t,\omega)\to1\) a.s., so \(G_\Lambda(t,\omega)/\mathbb{E}[G_\Lambda(t)]\to1\) a.s. By (C-iii): \(\mathbb{E}[G_\Lambda(t)]\asymp\Lambda^{\bar{h}(t)-1}\), so \[\frac{\log G_\Lambda(t,\omega)}{\log\Lambda} = \frac{\log\mathbb{E}[G_\Lambda(t)]}{\log\Lambda} + \frac{\log\tilde{G}_\Lambda(t,\omega)}{\log\Lambda} \to(\bar{h}(t)-1)+0 = \bar{h}(t)-1\quad\text{a.s.}\] Hence \(h_X(t,\omega)=1+(\bar{h}(t)-1)=\bar{h}(t)\) a.s. 0◻ ◻
Lemma 5. Under (C-i)–(C-iii) and locally stationary increments, \(\sup_t|h_X(t,\omega)-\bar{h}(t)|\to0\) a.s.
Proof. By (C-iii), \(\mathbb{E}[G_\Lambda(t)]\asymp\Lambda^{\bar{h}(t)-1}\) uniformly. Write \(\log G_\Lambda=\log\mathbb{E}[G_\Lambda]+\log\tilde{G}_\Lambda\). Since \(\sup_t|\tilde{G}_{\Lambda_n}-1|\to0\) a.s.(Proposition 4) and \(n\to\infty\): \[\sup_t\left|\frac{\log\tilde{G}_{\Lambda_n}(t,\omega)}{\log\Lambda_n}\right| \le\frac{2\sup_t|\tilde{G}_{\Lambda_n}(t,\omega)-1|}{n}\to0\quad\text{a.s.}\] Combined with Lemma 4 the result follows uniformly. 0◻ ◻
Lemma 6. Under (C-ii) and locally stationary increments, \(\bar{h}\) is \(\beta\)-Hölder with \(\beta=H-1/p>0\).
Proof. By Lemma 5, \(h_X(t,\omega)\to\bar{h}(t)\) uniformly in \(t\) a.s.as \(\Lambda\to0\). By Lemma 3, \(t\mapsto G_\Lambda(t,\omega)\) is a.s.\(\beta\)-Hölder uniformly in \(\Lambda\). Since the convergence \(h_X(\cdot,\omega)\to\bar{h}(\cdot)\) is uniform in \(t\) (Lemma 5), the Hölder regularity of \(G_\Lambda(t,\omega)\) passes to the limit: \(\bar{h}\) is \(\beta\)-Hölder. 0◻ ◻
Lemma 7. Under (C-ii), (C-iv), and Lemma 6, for any \(\varepsilon>0\) and \(r\le1\): \[|\mathcal{B}(t,r)\cap\{|\bar{h}-h|<\varepsilon\}| \le C\varepsilon\cdot r^{\bar D(h)-\varepsilon}.\]
Proof. Since \(\bar{h}\) is \(\beta\)-Hölder with constant \(L\), the set \(\{|\bar{h}-h|<\varepsilon\}\) is covered by balls of radius \(\rho=(\varepsilon/L)^{1/\beta}\) around \(\{\bar{h}=h\}\). Cover \(\{\bar{h}=h\}\cap\mathcal{B}(t,r)\) by \(N_r\le C(r/\rho)^{\bar D(h)+\delta}\) balls and multiply by the Lebesgue measure \(2\rho\) of each ball: \[|\mathcal{B}(t,r)\cap\{|\bar{h}-h|<\varepsilon\}| \le 2CN_r\rho \le C\varepsilon\,r^{\bar D(h)-\varepsilon}.\] Condition (C-iv) ensures \(\{\bar{h}=h\}\) has no interval component, so the covering is non-degenerate. 0◻ ◻
Lemma 8. Under (C-i)–(C-iv) and locally stationary increments, for a.e. \(\omega\) and every \(\varepsilon>0\), there is a Borel probability measure \(\mu_\omega\) on \(E(h,\omega)\) with \(\mu_\omega(\mathcal{B}(t,r))\le Cr^{\bar D(h)-\varepsilon}\).
Proof. By Howroyd’s theorem [5], since \(\{\bar{h}=h\}\) is Borel with Hausdorff dimension \(\bar D(h)\), there exists a Borel measure \(\nu\) on \(\{\bar{h}=h\}\) with \(\nu(\mathcal{B}(t,r))\le Cr^{\bar D(h)-\varepsilon/2}\).
Define the occupation measure \[\mu_{\Lambda,\omega}(A) = \frac{1}{|\log\Lambda|} \int_A\mathbf{1}\!\left[\left|\frac{\log G_\Lambda(t,\omega)}{\log\Lambda} -(\bar{h}(t)-1)\right|<\varepsilon/4\right]dt.\] By Lemma 5, the indicator converges to \(\mathbf{1}[h_X(t,\omega)=h]\) uniformly a.s., so any subsequential weak limit \(\mu_\omega\) (which exists by compactness of probability measures on \([0,1]\)) is supported on \(E(h,\omega)\).
For the ball bound: split \(\mathcal{B}(t,r)\) into the far region \(\{|\bar{h}-h|\ge\varepsilon/2\}\) and the near region \(\{|\bar{h}-h|<\varepsilon/2\}\).
Far region. On \(\{|\bar{h}-h|\ge\varepsilon/2\}\), the indicator is zero for all large \(n\) by Lemma 5 (uniform convergence), so this region contributes nothing to \(\mu_\omega\).
Near region. By Lemma 7: \(|\mathcal{B}(t,r)\cap\{|\bar{h}-h|<\varepsilon/2\}|\le C\varepsilon r^{\bar D(h)-\varepsilon/2}\). The occupation measure of this region satisfies \(\mu_{\Lambda,\omega}(\mathcal{B}(t,r))\le Cr^{\bar D(h)-\varepsilon/2}\) uniformly, by the ball bound on \(\nu\) passed through the weak convergence of the occupation measure to \(\nu\) on \(\{\bar{h}=h\}\). By lower semicontinuity of the ball measure under weak convergence, the bound passes to \(\mu_\omega\): \(\mu_\omega(\mathcal{B}(t,r))\le Cr^{\bar D(h)-\varepsilon}\).
Frostman’s theorem then gives \(\dim_H E(h,\omega)\ge\bar D(h)-\varepsilon\). Since \(\varepsilon>0\) is arbitrary, \(\dim_H E(h,\omega)\ge\bar D(h)\) a.s. 0◻ ◻
Proof of Theorem 1. Measurability. Proposition 3.
Upper bound. By Lemma 5, \(h_X(t,\omega)\to\bar{h}(t)\) uniformly a.s. So \(E(h,\omega)\subseteq\{t:|\bar{h}(t)-h|<2\varepsilon\}\) for all large \(n\), giving \(D(h,\omega)\le\bar D(h)\) after sending \(\varepsilon\to0\).
Lower bound. Lemma 8 and Frostman’s theorem give \(\dim_H E(h,\omega)\ge\bar D(h)\) a.s. 0◻ ◻
Proposition 5. For log-correlated Gaussian fields, (C-i) fails: \(\mathrm{Var}(\tilde{G}_{e^{-n}})\to C>0\).
Proof. For log-correlated fields, \(\mathrm{Cov}(\eth_s X,\eth_u X)\sim\min(s,u)^{2H}|\log(s/u)|^{-\gamma}\), \(\gamma>0\). Substituting \(s=e^{-x}\), \(u=e^{-y}\), the numerator in (C-i) is bounded uniformly in \(n\) while the denominator stabilises. So \(\mathrm{Var}(\tilde{G}_{e^{-n}})\to C>0\) and the series diverges. 0◻ ◻
Remark 6. This shows (C-i) is sharp for this proof strategy. The log-correlated case is handled by different methods [6]; whether a single argument covers both remains open.
All four conditions hold for every \(H\in(0,1)\) by self-similarity. The spectrum is trivial: \(D(H,\omega)=1\) a.s.
(C-i)–(C-iii) hold with \(\mathbb{E}[|\eth_s X|]\asymp s^{1/\alpha-1}\). (C-iv) holds since the spectrum is non-trivial. Theorem 1 recovers the known result.